I am now on the way to add the debias step to the whole algorithm. Just so that I have a set of “true parameters” for comparison, I am gonna use the simulation dataset for this part.
\[\begin{aligned} Y_i(t) & \sim Bernoulli(\frac{exp(\eta_i(t))}{1+exp(\eta_i(t))}) \\ \eta_i(t) &= f_0(t)+ \xi_{i1}\sqrt{2}sin(2\pi t)+\xi_{i2}\sqrt{2}cos(2\pi t)+\xi_{i3}\sqrt{2}sin(4\pi t)+\xi_{i4}\sqrt{2}cos(4\pi t) \end{aligned}\]
where:
N <- 500 # sample size
J <- 1000 # number of observation points
t = seq(0,1,len=J) # observations points
# mean function
f_0 <- function(s) 0
#eigenfunctions
K <- 4 # number of eigenfunctions
phi <- sqrt(2)*cbind(sin(2*pi*t),cos(2*pi*t),
sin(4*pi*t),cos(4*pi*t))
# eigenvalues
lambda = 0.5^(0:(K-1))
# generated training data
## score
xi <- matrix(rnorm(N*K),N,K)
xi <- xi %*% diag(sqrt(lambda))
# subject-specific random effect
b_i <- xi %*% t(phi); # of size N by J
# latent gaussian function
eta_i <- t(vapply(1:N, function(x){
f_0(t) + b_i[x,]
}, numeric(J)))
# outcome binary function
Y_i <- matrix(rbinom(N*J, size=1, prob=plogis(eta_i)),
N, J, byrow=FALSE)
# format into dataframe
# id = subject identifier (factor variable)
# sind = numeric value corresponding to the observed functional domain
# Y = functional response (binary)
# sind_inx = numeric value associated with order of "sind"
# this is not necessary, but may be of convenience when you implement the method
train_df <- data.frame(id = factor(rep(1:N, each=J)),
t = rep(t, N),
Y = as.vector(t(Y_i)),
eta_i = as.vector(t(eta_i)),
sind = rep(1:J, N))
# visualization
rand_id <- sample(N, size = 4)
train_df %>% filter(id %in% rand_id) %>%
ggplot()+
geom_point(aes(x=sind, y=Y), size = 0.5)+
geom_line(aes(x=sind, y=plogis(eta_i)), col = "red")+
#geom_line(aes(x=sind_inx, y=eta_i), col = "blue")+
facet_wrap(~id)
# bin data
bin_w <- 10 # bin width
n_bin <- J/bin_w # number of bins
brks <- seq(0, J, by = bin_w) # cutoff points
mid <- (brks+bin_w/2)[1:n_bin] # mid points
train_df$bin <- cut(train_df$sind, breaks = brks, include.lowest = T, labels = mid)
train_df$bin <- as.numeric(as.character(train_df$bin))
# unique(df$bin)
train_df %>%
filter(id %in% rand_id) %>%
group_by(id, bin) %>%
summarise(num = sum(Y)) %>%
ggplot()+
geom_point(aes(x=bin, y=num), size = 0.5)+
facet_wrap(~id)+
labs(x="Time", y = "Activity", title = "Number of active nimutes within each bin")
# fit model on the training set
train_bin_lst <- split(train_df, f = train_df$bin)
# length(train_bin_lst)
# lapply(train_bin_lst, dim)
# local GLMM and estimate latent function
# use PIRLS (nAGQ=0) to avoid near-unidentifiability issues
t1=Sys.time()
df_est_latent <- lapply(train_bin_lst, function(x){pred_latent(x, n_node = 0)})
t2= Sys.time()
t_local_glmm <- t2-t1 # less than 5 seconds
df_est_latent <- bind_rows(df_est_latent)
# head(df_est_latent)
# example estimated latent function
train_id <- unique(train_df$id)
rand_id <- sample(train_id, 4)
df_est_latent %>%
filter(id %in% rand_id) %>%
mutate(eta_hat = exp(eta_hat)/(1+exp(eta_hat))) %>%
mutate(eta_i = exp(eta_i)/(1+exp(eta_i))) %>%
ggplot()+
geom_line(aes(x=t, y=eta_hat, group = id, col = "estimated"))+
geom_line(aes(x=t, y=eta_i, group = id, col = "true"))+
geom_point(aes(x=t, y = Y, group = id), size = 0.5)+
facet_wrap(~id, scales = "free")+
labs(x = "Time", y = "Estimated latent function (probablity scale)")
uni_eta_hat <- df_est_latent %>% filter(bin==sind)
mid_t <- df_est_latent$t[df_est_latent$bin==df_est_latent$sind]
mid_t <- unique(mid_t) # the time points correspoinding to bin midpoints
mat_est_unique <- matrix(uni_eta_hat$eta_hat,
nrow=length(train_id),
ncol=n_bin, byrow = F)
# row index subject, column binned time
# dim(mat_est_unique)
t1 <- Sys.time()
fpca_mod <- fpca.face(mat_est_unique, argvals = mid_t, var=T)
t2 <- Sys.time()
t_fpca <- t2-t1 # less than one second
plot(mid_t, fpca_mod$mu, type = "l", xlab = "bin", ylab = "Mean")
par(mfrow=c(2,2))
plot(mid_t, fpca_mod$efunctions[, 1], type="l", xlab="bin", ylab="PC1")
plot(mid_t, fpca_mod$efunctions[, 2], type="l", xlab="bin", ylab="PC2")
plot(mid_t, fpca_mod$efunctions[, 3], type="l", xlab="bin", ylab="PC3")
plot(mid_t, fpca_mod$efunctions[, 4], type="l", xlab="bin", ylab="PC4")
# plot correlation matrix
heatmap(cov2cor(fpca_mod$VarMats[[1]]), Rowv = NA, Colv = NA, main = "Correlation")
# estimated eigenvalues from FPCA (biased)
fpca_mod$evalues[1:4]
## [1] 50.086753 23.167732 11.406167 6.059415
According to the fGFPCA paper, in this step we do two things:
# order of b splines
p <- 3
# knots
knots <- 35
knots_values <- seq(-p, knots + p, length = knots + 1 + 2 *p)/knots
knots_values <- knots_values * (max(mid_t) - min(mid_t)) + min(mid_t)
# evaluate B-splines on binned grid
B <- spline.des(knots = knots_values, x = mid_t, ord = p + 1,
outer.ok = TRUE)$design
# evaluate B-splines on original grid
Bnew <- spline.des(knots = knots_values, x = unique(train_df$t), ord = p + 1,
outer.ok = TRUE)$design
# project binned eigenfunctions onto the original grid
efunctions_new <- matrix(NA, J, K)
for(k in 1:K){
lm_mod <- lm(fpca_mod$efunctions[,k] ~ B-1)
efunctions_new[,k] <- Bnew %*% coef(lm_mod)
}
# project mean functions
# lm_mod <- lm(fpca_mod$mu ~ B-1)
# mu_new <- Bnew %*% coef(lm_mod)
# visualization
par(mfrow=c(2,2))
plot(mid_t, fpca_mod$efunctions[, 1], xlab="bin", ylab="PC1", pch=20, cex = 0.5)
lines(t, efunctions_new[, 1], col="blue")
plot(mid_t, fpca_mod$efunctions[, 2], xlab="bin", ylab="PC2", pch=20, cex = 0.5)
lines(t, efunctions_new[, 2], col="blue")
plot(mid_t, fpca_mod$efunctions[, 3], xlab="bin", ylab="PC3", pch=20, cex = 0.5)
lines(t, efunctions_new[, 3], col="blue")
plot(mid_t, fpca_mod$efunctions[, 4], xlab="bin", ylab="PC4", pch=20, cex = 0.5)
lines(t, efunctions_new[, 4], col="blue")
# dim(fpca_mod$efunctions)
df_phi <- data.frame(t = t, efunctions_new)
colnames(df_phi) <- c("t", paste0("phi", 1:4))
# train_df$bin <- as.numeric(as.character(train_df$bin))
train_df <- train_df %>% left_join(df_phi, by = "t")
train_df$id <- as.factor(train_df$id)
# usethis::edit_r_environ()
t1 <- Sys.time()
debias_glmm <- bam(Y ~ s(t, bs="cc", k=10)+
s(id, by=phi1, bs="re")+
s(id, by=phi2, bs="re")+
s(id, by=phi3, bs="re")+
s(id, by=phi4, bs="re"),
family = binomial, data=train_df,
method = "fREML",
discrete = TRUE)
t2 <- Sys.time()
t_debias <- t2-t1 # less than 45 seconds
We will need the mean function from this model, derived below:
The shape of the function depends very much on how the basis functions are constructed. For example, cubic spline would lead to a linear mean and cyclic cubic spline would lead to the function below. What’s a good way to decided?
new_mu <- predict(debias_glmm, type = "terms")
length(unique(new_mu[,1]))
## [1] 1000
new_mu <- unique(new_mu[,1])
plot(t, new_mu, type = "l")
And now I try to extract the variance of scores (debiased eigenvalues), using the conclusion that connects smoothing parameters to variance
\[\hat{p}_k=1/\hat{\lambda}_k^2\]
# transformed smoothing parameters
new_lambda <- 1/debias_glmm$sp[2:5]
new_lambda
## s(id):phi1 s(id):phi2 s(id):phi3 s(id):phi4
## 95.32322 47.95213 25.73439 13.18567
# new_lambda/lambda
# new_lambda/fpca_mod$evalues[1:4]
This eigenvalues, as well as the eigenfunctions, are actually not on the same scale of true values. By looking into the fpca.face source code, eigenfunctions are scaled by the square root of the number of total observations points. Since our model was fitted on the binnd grid, the actual eigenvaleus and eigenfunctions would be:
\[\begin{aligned} \phi_k(t) & = \sqrt{n_{bin}}\hat{\phi}_k(t) \\ \lambda_k & = \hat{\lambda}_k/n_{bin} \end{aligned}\]
Let’s rescale these quatities as below
scaled_lambda <- new_lambda/n_bin
efunctions_scaled <- efunctions_new*sqrt(n_bin)
par(mfrow=c(2,2))
plot(t, efunctions_scaled[, 1], type="l", xlab="bin", ylab="PC1")
plot(t, efunctions_scaled[, 2], type="l", xlab="bin", ylab="PC2")
plot(t, efunctions_scaled[, 3], type="l", xlab="bin", ylab="PC3")
plot(t, efunctions_scaled[, 4], type="l", xlab="bin", ylab="PC4")
I will now estimate individual score of test sample using Laplace Approximation, and calculated the predicted track.
# generated testing data
N_test <- 200
b_test <- matrix(rnorm(N_test*K), N_test, K) %*% diag(sqrt(lambda)) %*% t(phi) # of size N by J
# dim(b_test)
# latent gaussian function
eta_test <- t(vapply(1:N_test, function(x){
f_0(x) + b_i[x,]
}, numeric(J)))
# outcome binary function
Y_test <- matrix(rbinom(N_test*J, size=1, prob=plogis(eta_test)),
N_test, J, byrow=FALSE)
# format into dataframe
test_df <- data.frame(id = factor(rep(1:N_test, each=J)),
t = rep(t, N_test),
Y = as.vector(t(Y_test)),
eta_i = as.vector(t(eta_test)),
sind = rep(1:J, N_test))
# bin
test_df$bin <- cut(test_df$sind, breaks = brks, include.lowest = T, labels = mid)
test_df$bin <- as.numeric(as.character(test_df$bin))
test_df$id <- as.factor(test_df$id)
test_id <- unique(test_df$id)
# the model
## now, with expanded mean and eigenfuncitons, a Bernoulli distribution at each time point
Model <- function(parm, Data){
xi <- parm[Data$pos.xi]
# log-prior
xi.prior <- dmvnorm(xi, mean = rep(0, Data$K), sigma=Data$tao, log = TRUE)
# log-posterior likelihood
eta <- Data$f0+Data$X %*% xi
p <- exp(eta)/(1+exp(eta))
LL <- sum(dbern(x=Data$y, prob=p, log = TRUE)) # log likelihood of Y|xi
LP <- LL+sum(xi.prior) # joint log likelihood of (Y, xi)
# output
Modelout <- list(LP=LP, Dev=-2*LL, Monitor=LL,
yhat=Data$y, parm=parm)
return(Modelout)
}
# parameter names
name_lst <- as.list(rep(0, K))
names(name_lst) <- paste("xi", 1:K, sep = "")
parm.names <- as.parm.names(name_lst)
pos.xi <- grep("xi", parm.names)
PGF <- function(Data) {
xi <- rnorm(Data$K)
return(xi)
}
mon.names <- "LP"
# approximation process
# container
pred_list <- list()
# score <- array(NA, dim=c(length(test_id), K, 4))
# dimensions: subject, eigenfunction, maximum observation time
# prediction for a single subject
t1 <- Sys.time()
# per subject
for(i in seq_along(test_id)){
df_i <- test_df %>% filter(id==test_id[i])
# per max obs time
for(tmax in c(0.2, 0.4, 0.6, 0.8)){
df_it <- df_i %>% filter(t <= tmax)
max_rid <- nrow(df_it)
# into a list
MyData <- list(K=K, PGF=PGF,
X=efunctions_new[1:max_rid, ],
mon.names=mon.names,
parm.names=parm.names,
pos.xi=pos.xi,
y=df_it$Y,
tao=diag(new_lambda), f0=new_mu[1:max_rid])
# fit laplace approximation
Fit <- LaplaceApproximation(Model, parm = rep(0, K), Data=MyData, Method = "BFGS")
# Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values = parm = rep(0, K), Method = "BFGS")
score <- Fit$Summary1[, "Mode"]
# prediction
eta_pred_out <- new_mu+efunctions_new%*%score
df_i[ , paste0("pred", tmax)] <- eta_pred_out[,1]
}
# df_i$pred0.2[df_i$t<=0.2] <- NA
# df_i$pred0.4[df_i$t<=0.4] <- NA
# df_i$pred0.6[df_i$t<=0.6] <- NA
# df_i$pred0.8[df_i$t<=0.8] <- NA
pred_list[[i]] <- df_i
}
t2 <- Sys.time()
t_pred <- t2-t1 # About 3.5 minutes
df_pred <- bind_rows(pred_list)
df_pred$pred0.2[df_pred$t<=0.2] <- NA
df_pred$pred0.4[df_pred$t<=0.4] <- NA
df_pred$pred0.6[df_pred$t<=0.6] <- NA
df_pred$pred0.8[df_pred$t<=0.8] <- NA
rand_test_id <- sample(test_id, 4)
p1<-df_pred %>%
filter(id %in% rand_test_id) %>%
mutate_at(vars(eta_i, pred0.2, pred0.4, pred0.6, pred0.8), function(x){exp(x)/(1+exp(x))}) %>%
ggplot()+
geom_point(aes(x=t, y=Y), size = 0.2)+
geom_line(aes(x=t, y=eta_i, col = "True"))+
geom_line(aes(x=t, y=pred0.2, col = "0.2"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.4, col = "0.4"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.6, col = "0.6"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.8, col = "0.8"), linetype="dashed", na.rm=T)+
facet_wrap(~id)+
labs(title = "Unscaled, debiased eigenvalues")
## With debiased eigenvalues, also scale eigenvaleus and eigenfuncitons
# container
pred_list3 <- list()
# per subject
for(i in seq_along(test_id)){
df_i <- test_df %>% filter(id==test_id[i])
# per max obs time
for(tmax in c(0.2, 0.4, 0.6, 0.8)){
df_it <- df_i %>% filter(t <= tmax)
max_rid <- nrow(df_it)
# into a list
MyData <- list(K=K, PGF=PGF,
X=efunctions_scaled[1: max_rid, ],
mon.names=mon.names,
parm.names=parm.names,
pos.xi=pos.xi,
y=df_it$Y,
tao=diag(scaled_lambda), f0=new_mu[1:max_rid])
# fit laplace approximation
Fit <- LaplaceApproximation(Model, parm = rep(0, K), Data=MyData, Method = "BFGS")
score <- Fit$Summary1[, "Mode"]
# prediction
eta_pred_out <- new_mu+efunctions_scaled%*%score
df_i[ , paste0("pred", tmax)] <- eta_pred_out[,1]
}
# df_i$pred0.2[df_i$t<=0.2] <- NA
# df_i$pred0.4[df_i$t<=0.4] <- NA
# df_i$pred0.6[df_i$t<=0.6] <- NA
# df_i$pred0.8[df_i$t<=0.8] <- NA
pred_list3[[i]] <- df_i
}
df_pred3 <- bind_rows(pred_list3)
df_pred3$pred0.2[df_pred3$t<=0.2] <- NA
df_pred3$pred0.4[df_pred3$t<=0.4] <- NA
df_pred3$pred0.6[df_pred3$t<=0.6] <- NA
df_pred3$pred0.8[df_pred3$t<=0.8] <- NA
# overview of predicted track
p3 <- df_pred3 %>%
filter(id %in% rand_test_id) %>%
mutate_at(vars(eta_i, pred0.2, pred0.4, pred0.6, pred0.8), function(x){exp(x)/(1+exp(x))}) %>%
ggplot()+
geom_point(aes(x=t, y=Y), size = 0.2)+
geom_line(aes(x=t, y=eta_i, col = "True"))+
geom_line(aes(x=t, y=pred0.2, col = "0.2"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.4, col = "0.4"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.6, col = "0.6"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.8, col = "0.8"), linetype="dashed", na.rm=T)+
facet_wrap(~id)+
labs(title = "Scaled, unbiased eigenvalues")
t_1sim <- (t_local_glmm+t_fpca+t_debias+t_pred)
I would like to see if the following operations can improve predictive performance:
ggarrange(p1, p3, nrow = 1, common.legend = T)
# calcualte ISE
err1 <- df_pred %>%
mutate(err1 = (pred0.2-eta_i)^2,
err2 = (pred0.4-eta_i)^2,
err3 = (pred0.6-eta_i)^2,
err4 = (pred0.8-eta_i)^2) %>%
select(id, t, starts_with("err"))
err1$window = cut(err1$t, breaks = seq(0, 1, by = 0.2), include.lowest = T)
# table(err1$window)
err1 <- split(err1, f = err1$window)
tb1 <- lapply(err1, function(x){
x %>% group_by(id) %>%
summarise_at(vars(err1, err2, err3, err4), sum) %>%
summarise_at(vars(err1, err2, err3, err4), mean)
}) %>% bind_rows(.id = "Window")
# calcualte ISE
err3 <- df_pred3 %>%
mutate(err1 = (pred0.2-eta_i)^2,
err2 = (pred0.4-eta_i)^2,
err3 = (pred0.6-eta_i)^2,
err4 = (pred0.8-eta_i)^2) %>%
select(id, t, starts_with("err"))
err3$window = cut(err3$t, breaks = seq(0, 1, by = 0.2), include.lowest = T)
# table(err1$window)
err3 <- split(err3, f = err3$window)
tb3 <- lapply(err3, function(x){
x %>% group_by(id) %>%
summarise_at(vars(err1, err2, err3, err4), sum) %>%
summarise_at(vars(err1, err2, err3, err4), mean)
}) %>% bind_rows(.id = "Window")
# calcualte ISE
options(knitr.kable.NA = "")
ise <- full_join(tb1, tb3, by = "Window")
colnames(ise) <-c("Window", rep(seq(0.2, 0.8, by =0.2), 2))
ise %>%
kable(digits = 3, table.attr = "style = \"color: black;\"") %>%
kable_styling(full_width = F) %>%
add_header_above(c(" "= 1, "Unscaled" = 4, "Scaled" = 4)) %>%
add_header_above(c(" "=1, "Maximum observation time" = 8))
| Window | 0.2 | 0.4 | 0.6 | 0.8 | 0.2 | 0.4 | 0.6 | 0.8 |
|---|---|---|---|---|---|---|---|---|
| [0,0.2] | ||||||||
| (0.2,0.4] | 150.474 | 150.251 | ||||||
| (0.4,0.6] | 192.605 | 76.254 | 193.101 | 81.244 | ||||
| (0.6,0.8] | 224.855 | 53.395 | 18.161 | 224.123 | 56.692 | 18.338 | ||
| (0.8,1] | 118.223 | 88.065 | 22.398 | 13.259 | 118.133 | 95.631 | 22.768 | 13.318 |
# calcualte AUC
auc_df1 <- df_pred %>%
mutate(window = cut(t, breaks = seq(0, 1, by = 0.2), include.lowest = T)) %>%
select(Y, starts_with("pred"), window)
obs_track <- seq(0.2, 0.8, by = 0.2)
pred_window <- unique(auc_df1$window)
auc_mat <- matrix(NA, length(pred_window), length(obs_track))
for(j in seq_along(obs_track)){
for(i in (j+1):5){
this_Y <- auc_df1$Y[auc_df1$window==pred_window[i]]
this_pred <- auc_df1[auc_df1$window==pred_window[i],
paste0("pred", obs_track[j])]
this_perf <- performance(prediction(this_pred, this_Y), measure = "auc")
auc_mat[i ,j] <- this_perf@y.values[[1]]
}
}
# calcualte AUC
auc_df3 <- df_pred3 %>%
mutate(window = cut(t, breaks = seq(0, 1, by = 0.2), include.lowest = T)) %>%
select(Y, starts_with("pred"), window)
auc_mat3 <- matrix(NA, length(pred_window), length(obs_track))
for(j in seq_along(obs_track)){
for(i in (j+1):5){
this_Y <- auc_df3$Y[auc_df3$window==pred_window[i]]
this_pred <- auc_df3[auc_df3$window==pred_window[i],
paste0("pred", obs_track[j])]
this_perf <- performance(prediction(this_pred, this_Y), measure = "auc")
auc_mat3[i ,j] <- this_perf@y.values[[1]]
}
}
# calcualte AUC
options(knitr.kable.NA = "")
auc <- cbind(auc_mat, auc_mat3)
rownames(auc) <- unique(auc_df1$window)
colnames(auc) <-c(rep(seq(0.2, 0.8, by = 0.2), 2))
auc %>%
kable(digits = 3, table.attr = "style = \"color: black;\"") %>%
kable_styling(full_width = F) %>%
add_header_above(c(" "= 1, "Unscaled" = 4, "Scaled" = 4)) %>%
add_header_above(c(" "=1, "Maximum observation time" = 8))
| 0.2 | 0.4 | 0.6 | 0.8 | 0.2 | 0.4 | 0.6 | 0.8 | |
|---|---|---|---|---|---|---|---|---|
| [0,0.2] | ||||||||
| (0.2,0.4] | 0.737 | 0.737 | ||||||
| (0.4,0.6] | 0.666 | 0.744 | 0.666 | 0.743 | ||||
| (0.6,0.8] | 0.701 | 0.782 | 0.795 | 0.701 | 0.782 | 0.795 | ||
| (0.8,1] | 0.753 | 0.768 | 0.797 | 0.8 | 0.753 | 0.765 | 0.797 | 0.8 |
par(mfrow=c(2, 2))
plot(df_pred$pred0.2, df_pred3$pred0.2, pch=20, cex = 0.5,
xlab="unscaled", ylab="scaled", main = "Up to 0.2")
plot(df_pred$pred0.4, df_pred3$pred0.4, pch=20, cex = 0.5,
xlab="unscaled", ylab="scaled", main = "Up to 0.4")
plot(df_pred$pred0.6, df_pred3$pred0.6, pch=20, cex = 0.5,
xlab="unscaled", ylab="scaled", main = "Up to 0.6")
plot(df_pred$pred0.8, df_pred3$pred0.8, pch=20, cex = 0.5,
xlab="unscaled", ylab="scaled", main = "Up to 0.8")
data.frame(
df_pred$pred0.4[df_pred$t>0.4],
df_pred3$pred0.4[df_pred3$t>0.4]
)
## df_pred.pred0.4.df_pred.t...0.4. df_pred3.pred0.4.df_pred3.t...0.4.
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## 1596 1.593652e+00 1.435767e+00
## 1597 1.595136e+00 1.438056e+00
## 1598 1.596541e+00 1.440275e+00
## 1599 1.597866e+00 1.442424e+00
## 1600 1.599109e+00 1.444501e+00
## 1601 1.600271e+00 1.446505e+00
## 1602 1.601351e+00 1.448437e+00
## 1603 1.602348e+00 1.450295e+00
## 1604 1.603261e+00 1.452079e+00
## 1605 1.604090e+00 1.453788e+00
## 1606 1.604834e+00 1.455421e+00
## 1607 1.605493e+00 1.456977e+00
## 1608 1.606065e+00 1.458456e+00
## 1609 1.606551e+00 1.459857e+00
## 1610 1.606950e+00 1.461179e+00
## 1611 1.607261e+00 1.462421e+00
## 1612 1.607483e+00 1.463584e+00
## 1613 1.607616e+00 1.464665e+00
## 1614 1.607659e+00 1.465664e+00
## 1615 1.607612e+00 1.466581e+00
## 1616 1.607473e+00 1.467414e+00
## 1617 1.607243e+00 1.468164e+00
## 1618 1.606921e+00 1.468829e+00
## 1619 1.606505e+00 1.469408e+00
## 1620 1.605996e+00 1.469901e+00
## 1621 1.605393e+00 1.470307e+00
## 1622 1.604694e+00 1.470625e+00
## 1623 1.603900e+00 1.470855e+00
## 1624 1.603011e+00 1.470996e+00
## 1625 1.602024e+00 1.471046e+00
## 1626 1.600940e+00 1.471006e+00
## 1627 1.599758e+00 1.470875e+00
## 1628 1.598478e+00 1.470653e+00
## 1629 1.597100e+00 1.470339e+00
## 1630 1.595624e+00 1.469932e+00
## 1631 1.594049e+00 1.469433e+00
## 1632 1.592375e+00 1.468841e+00
## 1633 1.590603e+00 1.468156e+00
## 1634 1.588730e+00 1.467378e+00
## 1635 1.586759e+00 1.466506e+00
## 1636 1.584688e+00 1.465539e+00
## 1637 1.582516e+00 1.464479e+00
## 1638 1.580245e+00 1.463323e+00
## 1639 1.577873e+00 1.462072e+00
## 1640 1.575401e+00 1.460726e+00
## 1641 1.572827e+00 1.459283e+00
## 1642 1.570153e+00 1.457745e+00
## 1643 1.567377e+00 1.456110e+00
## 1644 1.564500e+00 1.454378e+00
## 1645 1.561522e+00 1.452549e+00
## 1646 1.558441e+00 1.450623e+00
## 1647 1.555258e+00 1.448598e+00
## 1648 1.551973e+00 1.446476e+00
## 1649 1.548585e+00 1.444255e+00
## 1650 1.545095e+00 1.441935e+00
## 1651 1.541501e+00 1.439515e+00
## 1652 1.537804e+00 1.436997e+00
## 1653 1.534004e+00 1.434378e+00
## 1654 1.530100e+00 1.431659e+00
## 1655 1.526092e+00 1.428840e+00
## 1656 1.521980e+00 1.425920e+00
## 1657 1.517765e+00 1.422900e+00
## 1658 1.513447e+00 1.419780e+00
## 1659 1.509026e+00 1.416560e+00
## 1660 1.504501e+00 1.413239e+00
## 1661 1.499874e+00 1.409818e+00
## 1662 1.495143e+00 1.406297e+00
## 1663 1.490310e+00 1.402676e+00
## 1664 1.485374e+00 1.398954e+00
## 1665 1.480336e+00 1.395133e+00
## 1666 1.475196e+00 1.391212e+00
## 1667 1.469953e+00 1.387190e+00
## 1668 1.464609e+00 1.383069e+00
## 1669 1.459162e+00 1.378847e+00
## 1670 1.453614e+00 1.374526e+00
## 1671 1.447965e+00 1.370105e+00
## 1672 1.442213e+00 1.365584e+00
## 1673 1.436361e+00 1.360963e+00
## 1674 1.430407e+00 1.356243e+00
## 1675 1.424352e+00 1.351423e+00
## 1676 1.418196e+00 1.346503e+00
## 1677 1.411939e+00 1.341483e+00
## 1678 1.405582e+00 1.336364e+00
## 1679 1.399124e+00 1.331145e+00
## 1680 1.392566e+00 1.325827e+00
## 1681 1.385907e+00 1.320409e+00
## 1682 1.379148e+00 1.314891e+00
## 1683 1.372290e+00 1.309274e+00
## 1684 1.365332e+00 1.303559e+00
## 1685 1.358275e+00 1.297745e+00
## 1686 1.351122e+00 1.291835e+00
## 1687 1.343871e+00 1.285827e+00
## 1688 1.336524e+00 1.279724e+00
## 1689 1.329081e+00 1.273525e+00
## 1690 1.321544e+00 1.267232e+00
## 1691 1.313913e+00 1.260845e+00
## 1692 1.306189e+00 1.254365e+00
## 1693 1.298373e+00 1.247792e+00
## 1694 1.290465e+00 1.241127e+00
## 1695 1.282466e+00 1.234372e+00
## 1696 1.274376e+00 1.227525e+00
## 1697 1.266198e+00 1.220589e+00
## 1698 1.257931e+00 1.213564e+00
## 1699 1.249576e+00 1.206451e+00
## 1700 1.241133e+00 1.199250e+00
## 1701 1.232605e+00 1.191962e+00
## 1702 1.223990e+00 1.184587e+00
## 1703 1.215291e+00 1.177127e+00
## 1704 1.206508e+00 1.169582e+00
## 1705 1.197642e+00 1.161953e+00
## 1706 1.188692e+00 1.154240e+00
## 1707 1.179661e+00 1.146444e+00
## 1708 1.170549e+00 1.138566e+00
## 1709 1.161356e+00 1.130606e+00
## 1710 1.152084e+00 1.122566e+00
## 1711 1.142733e+00 1.114445e+00
## 1712 1.133305e+00 1.106246e+00
## 1713 1.123799e+00 1.097968e+00
## 1714 1.114218e+00 1.089613e+00
## 1715 1.104563e+00 1.081183e+00
## 1716 1.094834e+00 1.072678e+00
## 1717 1.085033e+00 1.064099e+00
## 1718 1.075161e+00 1.055448e+00
## 1719 1.065218e+00 1.046725e+00
## 1720 1.055207e+00 1.037932e+00
## 1721 1.045128e+00 1.029069e+00
## 1722 1.034983e+00 1.020139e+00
## 1723 1.024772e+00 1.011141e+00
## 1724 1.014496e+00 1.002077e+00
## 1725 1.004157e+00 9.929489e-01
## 1726 9.937560e-01 9.837566e-01
## 1727 9.832937e-01 9.745015e-01
## 1728 9.727714e-01 9.651849e-01
## 1729 9.621902e-01 9.558077e-01
## 1730 9.515512e-01 9.463711e-01
## 1731 9.408556e-01 9.368762e-01
## 1732 9.301043e-01 9.273241e-01
## 1733 9.192987e-01 9.177158e-01
## 1734 9.084396e-01 9.080525e-01
## 1735 8.975284e-01 8.983352e-01
## 1736 8.865660e-01 8.885651e-01
## 1737 8.755536e-01 8.787432e-01
## 1738 8.644923e-01 8.688706e-01
## 1739 8.533833e-01 8.589485e-01
## 1740 8.422275e-01 8.489779e-01
## 1741 8.310260e-01 8.389598e-01
## 1742 8.197799e-01 8.288954e-01
## 1743 8.084903e-01 8.187857e-01
## 1744 7.971581e-01 8.086318e-01
## 1745 7.857844e-01 7.984348e-01
## 1746 7.743702e-01 7.881957e-01
## 1747 7.629166e-01 7.779156e-01
## 1748 7.514247e-01 7.675955e-01
## 1749 7.398954e-01 7.572366e-01
## 1750 7.283298e-01 7.468399e-01
## 1751 7.167290e-01 7.364065e-01
## 1752 7.050940e-01 7.259374e-01
## 1753 6.934258e-01 7.154337e-01
## 1754 6.817255e-01 7.048965e-01
## 1755 6.699941e-01 6.943269e-01
## 1756 6.582327e-01 6.837259e-01
## 1757 6.464423e-01 6.730946e-01
## 1758 6.346240e-01 6.624341e-01
## 1759 6.227788e-01 6.517454e-01
## 1760 6.109077e-01 6.410296e-01
## 1761 5.990118e-01 6.302878e-01
## 1762 5.870921e-01 6.195211e-01
## 1763 5.751496e-01 6.087304e-01
## 1764 5.631855e-01 5.979170e-01
## 1765 5.512008e-01 5.870818e-01
## 1766 5.391964e-01 5.762260e-01
## 1767 5.271735e-01 5.653505e-01
## 1768 5.151328e-01 5.544563e-01
## 1769 5.030751e-01 5.435441e-01
## 1770 4.910008e-01 5.326143e-01
## 1771 4.789104e-01 5.216676e-01
## 1772 4.668046e-01 5.107045e-01
## 1773 4.546839e-01 4.997256e-01
## 1774 4.425487e-01 4.887315e-01
## 1775 4.303997e-01 4.777227e-01
## 1776 4.182374e-01 4.666999e-01
## 1777 4.060624e-01 4.556636e-01
## 1778 3.938751e-01 4.446144e-01
## 1779 3.816762e-01 4.335528e-01
## 1780 3.694661e-01 4.224795e-01
## 1781 3.572455e-01 4.113950e-01
## 1782 3.450149e-01 4.002998e-01
## 1783 3.327747e-01 3.891947e-01
## 1784 3.205256e-01 3.780800e-01
## 1785 3.082682e-01 3.669565e-01
## 1786 2.960029e-01 3.558247e-01
## 1787 2.837302e-01 3.446851e-01
## 1788 2.714509e-01 3.335384e-01
## 1789 2.591653e-01 3.223851e-01
## 1790 2.468740e-01 3.112258e-01
## 1791 2.345776e-01 3.000611e-01
## 1792 2.222767e-01 2.888915e-01
## 1793 2.099717e-01 2.777176e-01
## 1794 1.976632e-01 2.665400e-01
## 1795 1.853518e-01 2.553593e-01
## 1796 1.730417e-01 2.441787e-01
## 1797 1.607519e-01 2.330125e-01
## 1798 1.485052e-01 2.218773e-01
## 1799 1.363244e-01 2.107901e-01
## 1800 1.242322e-01 1.997677e-01
## 1801 -6.506134e-02 -6.503087e-02
## 1802 -5.826280e-02 -5.823211e-02
## 1803 -5.142213e-02 -5.139124e-02
## 1804 -4.453961e-02 -4.450851e-02
## 1805 -3.761549e-02 -3.758418e-02
## 1806 -3.065003e-02 -3.061852e-02
## 1807 -2.364348e-02 -2.361177e-02
## 1808 -1.659611e-02 -1.656421e-02
## 1809 -9.508168e-03 -9.476077e-03
## 1810 -2.379919e-03 -2.347642e-03
## 1811 4.788382e-03 4.820841e-03
## 1812 1.199647e-02 1.202911e-02
## 1813 1.924410e-02 1.927691e-02
## 1814 2.653101e-02 2.656399e-02
## 1815 3.385693e-02 3.389008e-02
## 1816 4.122161e-02 4.125492e-02
## 1817 4.862479e-02 4.865827e-02
## 1818 5.606622e-02 5.609985e-02
## 1819 6.354563e-02 6.357941e-02
## 1820 7.106277e-02 7.109670e-02
## 1821 7.861738e-02 7.865145e-02
## 1822 8.620920e-02 8.624341e-02
## 1823 9.383797e-02 9.387231e-02
## 1824 1.015034e-01 1.015379e-01
## 1825 1.092053e-01 1.092399e-01
## 1826 1.169434e-01 1.169781e-01
## 1827 1.247174e-01 1.247523e-01
## 1828 1.325271e-01 1.325620e-01
## 1829 1.403721e-01 1.404072e-01
## 1830 1.482523e-01 1.482875e-01
## 1831 1.561672e-01 1.562025e-01
## 1832 1.641164e-01 1.641517e-01
## 1833 1.720992e-01 1.721347e-01
## 1834 1.801154e-01 1.801509e-01
## 1835 1.881642e-01 1.881998e-01
## 1836 1.962453e-01 1.962810e-01
## 1837 2.043582e-01 2.043940e-01
## 1838 2.125023e-01 2.125382e-01
## 1839 2.206773e-01 2.207132e-01
## 1840 2.288825e-01 2.289185e-01
## 1841 2.371175e-01 2.371535e-01
## 1842 2.453818e-01 2.454179e-01
## 1843 2.536749e-01 2.537110e-01
## 1844 2.619963e-01 2.620325e-01
## 1845 2.703455e-01 2.703817e-01
## 1846 2.787221e-01 2.787583e-01
## 1847 2.871255e-01 2.871618e-01
## 1848 2.955552e-01 2.955915e-01
## 1849 3.040108e-01 3.040471e-01
## 1850 3.124917e-01 3.125281e-01
## 1851 3.209975e-01 3.210339e-01
## 1852 3.295277e-01 3.295641e-01
## 1853 3.380818e-01 3.381182e-01
## 1854 3.466592e-01 3.466956e-01
## 1855 3.552595e-01 3.552959e-01
## 1856 3.638823e-01 3.639187e-01
## 1857 3.725269e-01 3.725633e-01
## 1858 3.811930e-01 3.812294e-01
## 1859 3.898799e-01 3.899162e-01
## 1860 3.985870e-01 3.986233e-01
## 1861 4.073135e-01 4.073497e-01
## 1862 4.160588e-01 4.160950e-01
## 1863 4.248221e-01 4.248583e-01
## 1864 4.336029e-01 4.336391e-01
## 1865 4.424004e-01 4.424365e-01
## 1866 4.512139e-01 4.512499e-01
## 1867 4.600427e-01 4.600787e-01
## 1868 4.688862e-01 4.689221e-01
## 1869 4.777436e-01 4.777795e-01
## 1870 4.866143e-01 4.866501e-01
## 1871 4.954976e-01 4.955333e-01
## 1872 5.043928e-01 5.044284e-01
## 1873 5.132992e-01 5.133347e-01
## 1874 5.222160e-01 5.222515e-01
## 1875 5.311428e-01 5.311781e-01
## 1876 5.400786e-01 5.401138e-01
## 1877 5.490229e-01 5.490580e-01
## 1878 5.579749e-01 5.580100e-01
## 1879 5.669340e-01 5.669690e-01
## 1880 5.758995e-01 5.759343e-01
## 1881 5.848707e-01 5.849054e-01
## 1882 5.938469e-01 5.938815e-01
## 1883 6.028274e-01 6.028619e-01
## 1884 6.118116e-01 6.118459e-01
## 1885 6.207986e-01 6.208328e-01
## 1886 6.297880e-01 6.298220e-01
## 1887 6.387789e-01 6.388127e-01
## 1888 6.477707e-01 6.478044e-01
## 1889 6.567629e-01 6.567965e-01
## 1890 6.657548e-01 6.657882e-01
## 1891 6.747459e-01 6.747791e-01
## 1892 6.837354e-01 6.837684e-01
## 1893 6.927228e-01 6.927556e-01
## 1894 7.017074e-01 7.017401e-01
## 1895 7.106886e-01 7.107212e-01
## 1896 7.196659e-01 7.196983e-01
## 1897 7.286386e-01 7.286708e-01
## 1898 7.376061e-01 7.376380e-01
## 1899 7.465677e-01 7.465995e-01
## 1900 7.555229e-01 7.555545e-01
## 1901 7.644710e-01 7.645024e-01
## 1902 7.734115e-01 7.734426e-01
## 1903 7.823436e-01 7.823745e-01
## 1904 7.912668e-01 7.912975e-01
## 1905 8.001805e-01 8.002110e-01
## 1906 8.090840e-01 8.091143e-01
## 1907 8.179768e-01 8.180068e-01
## 1908 8.268582e-01 8.268880e-01
## 1909 8.357275e-01 8.357571e-01
## 1910 8.445843e-01 8.446136e-01
## 1911 8.534278e-01 8.534569e-01
## 1912 8.622575e-01 8.622863e-01
## 1913 8.710727e-01 8.711013e-01
## 1914 8.798728e-01 8.799011e-01
## 1915 8.886572e-01 8.886852e-01
## 1916 8.974250e-01 8.974528e-01
## 1917 9.061753e-01 9.062029e-01
## 1918 9.149074e-01 9.149346e-01
## 1919 9.236201e-01 9.236472e-01
## 1920 9.323128e-01 9.323395e-01
## 1921 9.409845e-01 9.410109e-01
## 1922 9.496342e-01 9.496604e-01
## 1923 9.582611e-01 9.582870e-01
## 1924 9.668644e-01 9.668900e-01
## 1925 9.754430e-01 9.754683e-01
## 1926 9.839962e-01 9.840212e-01
## 1927 9.925230e-01 9.925477e-01
## 1928 1.001023e+00 1.001047e+00
## 1929 1.009494e+00 1.009518e+00
## 1930 1.017936e+00 1.017960e+00
## 1931 1.026349e+00 1.026372e+00
## 1932 1.034730e+00 1.034753e+00
## 1933 1.043080e+00 1.043103e+00
## 1934 1.051397e+00 1.051420e+00
## 1935 1.059681e+00 1.059703e+00
## 1936 1.067931e+00 1.067952e+00
## 1937 1.076145e+00 1.076166e+00
## 1938 1.084322e+00 1.084344e+00
## 1939 1.092463e+00 1.092484e+00
## 1940 1.100566e+00 1.100587e+00
## 1941 1.108630e+00 1.108650e+00
## 1942 1.116654e+00 1.116674e+00
## 1943 1.124638e+00 1.124657e+00
## 1944 1.132580e+00 1.132599e+00
## 1945 1.140479e+00 1.140498e+00
## 1946 1.148334e+00 1.148353e+00
## 1947 1.156145e+00 1.156163e+00
## 1948 1.163909e+00 1.163927e+00
## 1949 1.171627e+00 1.171644e+00
## 1950 1.179296e+00 1.179313e+00
## 1951 1.186916e+00 1.186933e+00
## 1952 1.194485e+00 1.194502e+00
## 1953 1.202004e+00 1.202020e+00
## 1954 1.209469e+00 1.209485e+00
## 1955 1.216881e+00 1.216896e+00
## 1956 1.224239e+00 1.224253e+00
## 1957 1.231540e+00 1.231555e+00
## 1958 1.238785e+00 1.238799e+00
## 1959 1.245971e+00 1.245985e+00
## 1960 1.253099e+00 1.253112e+00
## 1961 1.260167e+00 1.260179e+00
## 1962 1.267173e+00 1.267185e+00
## 1963 1.274117e+00 1.274129e+00
## 1964 1.280997e+00 1.281009e+00
## 1965 1.287813e+00 1.287825e+00
## 1966 1.294564e+00 1.294575e+00
## 1967 1.301247e+00 1.301258e+00
## 1968 1.307863e+00 1.307873e+00
## 1969 1.314410e+00 1.314420e+00
## 1970 1.320888e+00 1.320897e+00
## 1971 1.327294e+00 1.327303e+00
## 1972 1.333628e+00 1.333636e+00
## 1973 1.339889e+00 1.339897e+00
## 1974 1.346077e+00 1.346084e+00
## 1975 1.352190e+00 1.352197e+00
## 1976 1.358227e+00 1.358234e+00
## 1977 1.364189e+00 1.364195e+00
## 1978 1.370073e+00 1.370079e+00
## 1979 1.375879e+00 1.375885e+00
## 1980 1.381607e+00 1.381612e+00
## 1981 1.387255e+00 1.387260e+00
## 1982 1.392823e+00 1.392827e+00
## 1983 1.398309e+00 1.398313e+00
## 1984 1.403714e+00 1.403717e+00
## 1985 1.409035e+00 1.409038e+00
## 1986 1.414273e+00 1.414275e+00
## 1987 1.419426e+00 1.419428e+00
## 1988 1.424493e+00 1.424495e+00
## 1989 1.429475e+00 1.429476e+00
## 1990 1.434369e+00 1.434370e+00
## 1991 1.439175e+00 1.439176e+00
## 1992 1.443893e+00 1.443893e+00
## 1993 1.448521e+00 1.448520e+00
## 1994 1.453058e+00 1.453058e+00
## 1995 1.457504e+00 1.457503e+00
## 1996 1.461859e+00 1.461857e+00
## 1997 1.466120e+00 1.466118e+00
## 1998 1.470287e+00 1.470285e+00
## 1999 1.474360e+00 1.474357e+00
## 2000 1.478337e+00 1.478334e+00
## 2001 1.482218e+00 1.482214e+00
## 2002 1.486002e+00 1.485998e+00
## 2003 1.489689e+00 1.489685e+00
## 2004 1.493278e+00 1.493273e+00
## 2005 1.496769e+00 1.496763e+00
## 2006 1.500160e+00 1.500154e+00
## 2007 1.503451e+00 1.503444e+00
## 2008 1.506641e+00 1.506634e+00
## 2009 1.509730e+00 1.509723e+00
## 2010 1.512717e+00 1.512710e+00
## 2011 1.515602e+00 1.515594e+00
## 2012 1.518383e+00 1.518375e+00
## 2013 1.521061e+00 1.521052e+00
## 2014 1.523634e+00 1.523625e+00
## 2015 1.526102e+00 1.526092e+00
## 2016 1.528464e+00 1.528454e+00
## 2017 1.530720e+00 1.530709e+00
## 2018 1.532869e+00 1.532858e+00
## 2019 1.534910e+00 1.534898e+00
## 2020 1.536842e+00 1.536831e+00
## 2021 1.538666e+00 1.538654e+00
## 2022 1.540380e+00 1.540368e+00
## 2023 1.541984e+00 1.541971e+00
## 2024 1.543477e+00 1.543464e+00
## 2025 1.544858e+00 1.544844e+00
## 2026 1.546127e+00 1.546113e+00
## 2027 1.547284e+00 1.547269e+00
## 2028 1.548326e+00 1.548312e+00
## 2029 1.549255e+00 1.549240e+00
## 2030 1.550070e+00 1.550055e+00
## 2031 1.550771e+00 1.550755e+00
## 2032 1.551357e+00 1.551340e+00
## 2033 1.551828e+00 1.551811e+00
## 2034 1.552184e+00 1.552167e+00
## 2035 1.552425e+00 1.552407e+00
## 2036 1.552550e+00 1.552532e+00
## 2037 1.552560e+00 1.552542e+00
## 2038 1.552454e+00 1.552435e+00
## 2039 1.552231e+00 1.552212e+00
## 2040 1.551892e+00 1.551872e+00
## 2041 1.551436e+00 1.551416e+00
## 2042 1.550863e+00 1.550843e+00
## 2043 1.550173e+00 1.550152e+00
## 2044 1.549366e+00 1.549345e+00
## 2045 1.548440e+00 1.548419e+00
## 2046 1.547397e+00 1.547376e+00
## 2047 1.546236e+00 1.546214e+00
## 2048 1.544956e+00 1.544934e+00
## 2049 1.543558e+00 1.543535e+00
## 2050 1.542041e+00 1.542018e+00
## 2051 1.540405e+00 1.540381e+00
## 2052 1.538649e+00 1.538625e+00
## 2053 1.536774e+00 1.536750e+00
## 2054 1.534779e+00 1.534754e+00
## 2055 1.532663e+00 1.532639e+00
## 2056 1.530428e+00 1.530403e+00
## 2057 1.528072e+00 1.528047e+00
## 2058 1.525597e+00 1.525571e+00
## 2059 1.523001e+00 1.522975e+00
## 2060 1.520287e+00 1.520260e+00
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## 2064 1.508245e+00 1.508217e+00
## 2065 1.504940e+00 1.504912e+00
## 2066 1.501518e+00 1.501490e+00
## 2067 1.497980e+00 1.497951e+00
## 2068 1.494325e+00 1.494296e+00
## 2069 1.490555e+00 1.490525e+00
## 2070 1.486669e+00 1.486639e+00
## 2071 1.482668e+00 1.482638e+00
## 2072 1.478552e+00 1.478522e+00
## 2073 1.474321e+00 1.474291e+00
## 2074 1.469977e+00 1.469947e+00
## 2075 1.465519e+00 1.465489e+00
## 2076 1.460948e+00 1.460917e+00
## 2077 1.456264e+00 1.456233e+00
## 2078 1.451468e+00 1.451436e+00
## 2079 1.446559e+00 1.446527e+00
## 2080 1.441538e+00 1.441506e+00
## 2081 1.436406e+00 1.436374e+00
## 2082 1.431164e+00 1.431131e+00
## 2083 1.425810e+00 1.425777e+00
## 2084 1.420346e+00 1.420313e+00
## 2085 1.414772e+00 1.414739e+00
## 2086 1.409089e+00 1.409056e+00
## 2087 1.403298e+00 1.403264e+00
## 2088 1.397398e+00 1.397365e+00
## 2089 1.391392e+00 1.391358e+00
## 2090 1.385279e+00 1.385245e+00
## 2091 1.379061e+00 1.379026e+00
## 2092 1.372737e+00 1.372703e+00
## 2093 1.366310e+00 1.366275e+00
## 2094 1.359778e+00 1.359743e+00
## 2095 1.353144e+00 1.353109e+00
## 2096 1.346408e+00 1.346372e+00
## 2097 1.339570e+00 1.339534e+00
## 2098 1.332631e+00 1.332596e+00
## 2099 1.325592e+00 1.325557e+00
## 2100 1.318454e+00 1.318418e+00
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## 2102 1.303883e+00 1.303847e+00
## 2103 1.296451e+00 1.296414e+00
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## 2106 1.273578e+00 1.273541e+00
## 2107 1.265764e+00 1.265727e+00
## 2108 1.257856e+00 1.257819e+00
## 2109 1.249855e+00 1.249818e+00
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## 19883 -2.288523e-01 -2.288041e-01
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## 19888 -2.083847e-01 -2.083318e-01
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## 19890 -2.000512e-01 -1.999965e-01
## 19891 -1.958523e-01 -1.957967e-01
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## 19893 -1.873891e-01 -1.873317e-01
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## 19902 -1.481851e-01 -1.481197e-01
## 19903 -1.437120e-01 -1.436457e-01
## 19904 -1.392147e-01 -1.391476e-01
## 19905 -1.346930e-01 -1.346250e-01
## 19906 -1.301467e-01 -1.300779e-01
## 19907 -1.255756e-01 -1.255060e-01
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## 19909 -1.163584e-01 -1.162870e-01
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## 19911 -1.070395e-01 -1.069665e-01
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## 19935 1.316914e-02 1.326014e-02
## 19936 1.852733e-02 1.861898e-02
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## 32003 2.474301e+00 2.476472e+00
## 32004 2.488523e+00 2.490695e+00
## 32005 2.502650e+00 2.504823e+00
## 32006 2.516681e+00 2.518856e+00
## 32007 2.530617e+00 2.532792e+00
## 32008 2.544454e+00 2.546631e+00
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## 32010 2.571832e+00 2.574010e+00
## 32011 2.585370e+00 2.587548e+00
## 32012 2.598806e+00 2.600984e+00
## 32013 2.612139e+00 2.614317e+00
## 32014 2.625368e+00 2.627545e+00
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## 32016 2.651508e+00 2.653683e+00
## 32017 2.664417e+00 2.666591e+00
## 32018 2.677217e+00 2.679390e+00
## 32019 2.689908e+00 2.692079e+00
## 32020 2.702488e+00 2.704657e+00
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## 32052 3.040750e+00 3.042736e+00
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## 40922 8.206917e-01 8.196955e-01
## 40923 8.513652e-01 8.503751e-01
## 40924 8.819729e-01 8.809889e-01
## 40925 9.125113e-01 9.115333e-01
## 40926 9.429766e-01 9.420049e-01
## 40927 9.733654e-01 9.723999e-01
## 40928 1.003674e+00 1.002715e+00
## 40929 1.033899e+00 1.032946e+00
## 40930 1.064036e+00 1.063089e+00
## 40931 1.094082e+00 1.093142e+00
## 40932 1.124034e+00 1.123100e+00
## 40933 1.153887e+00 1.152960e+00
## 40934 1.183638e+00 1.182718e+00
## 40935 1.213284e+00 1.212370e+00
## 40936 1.242821e+00 1.241914e+00
## 40937 1.272245e+00 1.271345e+00
## 40938 1.301553e+00 1.300659e+00
## 40939 1.330741e+00 1.329854e+00
## 40940 1.359805e+00 1.358925e+00
## 40941 1.388742e+00 1.387869e+00
## 40942 1.417548e+00 1.416682e+00
## 40943 1.446220e+00 1.445361e+00
## 40944 1.474754e+00 1.473902e+00
## 40945 1.503146e+00 1.502301e+00
## 40946 1.531394e+00 1.530555e+00
## 40947 1.559492e+00 1.558661e+00
## 40948 1.587438e+00 1.586614e+00
## 40949 1.615227e+00 1.614411e+00
## 40950 1.642857e+00 1.642048e+00
## 40951 1.670324e+00 1.669522e+00
## 40952 1.697624e+00 1.696829e+00
## 40953 1.724753e+00 1.723966e+00
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## 40957 1.831494e+00 1.830736e+00
## 40958 1.857716e+00 1.856966e+00
## 40959 1.883747e+00 1.883004e+00
## 40960 1.909581e+00 1.908847e+00
## 40961 1.935217e+00 1.934490e+00
## 40962 1.960649e+00 1.959929e+00
## 40963 1.985874e+00 1.985163e+00
## 40964 2.010890e+00 2.010186e+00
## 40965 2.035691e+00 2.034995e+00
## 40966 2.060275e+00 2.059587e+00
## 40967 2.084638e+00 2.083958e+00
## 40968 2.108777e+00 2.108104e+00
## 40969 2.132687e+00 2.132022e+00
## 40970 2.156365e+00 2.155708e+00
## 40971 2.179807e+00 2.179158e+00
## 40972 2.203011e+00 2.202370e+00
## 40973 2.225972e+00 2.225339e+00
## 40974 2.248689e+00 2.248064e+00
## 40975 2.271158e+00 2.270541e+00
## 40976 2.293375e+00 2.292767e+00
## 40977 2.315339e+00 2.314739e+00
## 40978 2.337046e+00 2.336453e+00
## 40979 2.358493e+00 2.357908e+00
## 40980 2.379677e+00 2.379101e+00
## 40981 2.400595e+00 2.400027e+00
## 40982 2.421244e+00 2.420684e+00
## 40983 2.441621e+00 2.441069e+00
## 40984 2.461723e+00 2.461180e+00
## 40985 2.481547e+00 2.481012e+00
## 40986 2.501091e+00 2.500564e+00
## 40987 2.520350e+00 2.519832e+00
## 40988 2.539323e+00 2.538813e+00
## 40989 2.558005e+00 2.557503e+00
## 40990 2.576395e+00 2.575901e+00
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## 40992 2.612284e+00 2.611807e+00
## 40993 2.629777e+00 2.629308e+00
## 40994 2.646965e+00 2.646504e+00
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## 49790 9.996604e-01 9.996209e-01
## 49791 1.001320e+00 1.001281e+00
## 49792 1.002978e+00 1.002940e+00
## 49793 1.004634e+00 1.004596e+00
## 49794 1.006288e+00 1.006251e+00
## 49795 1.007941e+00 1.007904e+00
## 49796 1.009585e+00 1.009548e+00
## 49797 1.011179e+00 1.011143e+00
## 49798 1.012674e+00 1.012639e+00
## 49799 1.014023e+00 1.013988e+00
## 49800 1.015175e+00 1.015141e+00
## 49801 -1.298594e+00 -1.298519e+00
## 49802 -1.301203e+00 -1.301128e+00
## 49803 -1.303633e+00 -1.303558e+00
## 49804 -1.305884e+00 -1.305810e+00
## 49805 -1.307955e+00 -1.307882e+00
## 49806 -1.309846e+00 -1.309774e+00
## 49807 -1.311557e+00 -1.311485e+00
## 49808 -1.313087e+00 -1.313015e+00
## 49809 -1.314434e+00 -1.314363e+00
## 49810 -1.315600e+00 -1.315530e+00
## 49811 -1.316583e+00 -1.316513e+00
## 49812 -1.317382e+00 -1.317313e+00
## 49813 -1.317997e+00 -1.317929e+00
## 49814 -1.318428e+00 -1.318360e+00
## 49815 -1.318674e+00 -1.318606e+00
## 49816 -1.318734e+00 -1.318667e+00
## 49817 -1.318608e+00 -1.318542e+00
## 49818 -1.318295e+00 -1.318230e+00
## 49819 -1.317795e+00 -1.317730e+00
## 49820 -1.317107e+00 -1.317043e+00
## 49821 -1.316231e+00 -1.316167e+00
## 49822 -1.315166e+00 -1.315103e+00
## 49823 -1.313911e+00 -1.313849e+00
## 49824 -1.312466e+00 -1.312405e+00
## 49825 -1.310831e+00 -1.310770e+00
## 49826 -1.309004e+00 -1.308944e+00
## 49827 -1.306986e+00 -1.306927e+00
## 49828 -1.304776e+00 -1.304717e+00
## 49829 -1.302372e+00 -1.302314e+00
## 49830 -1.299775e+00 -1.299718e+00
## 49831 -1.296985e+00 -1.296929e+00
## 49832 -1.294002e+00 -1.293946e+00
## 49833 -1.290826e+00 -1.290771e+00
## 49834 -1.287457e+00 -1.287402e+00
## 49835 -1.283895e+00 -1.283842e+00
## 49836 -1.280141e+00 -1.280088e+00
## 49837 -1.276195e+00 -1.276143e+00
## 49838 -1.272057e+00 -1.272006e+00
## 49839 -1.267728e+00 -1.267677e+00
## 49840 -1.263207e+00 -1.263157e+00
## 49841 -1.258494e+00 -1.258445e+00
## 49842 -1.253591e+00 -1.253542e+00
## 49843 -1.248496e+00 -1.248449e+00
## 49844 -1.243211e+00 -1.243164e+00
## 49845 -1.237736e+00 -1.237690e+00
## 49846 -1.232070e+00 -1.232025e+00
## 49847 -1.226214e+00 -1.226170e+00
## 49848 -1.220169e+00 -1.220125e+00
## 49849 -1.213934e+00 -1.213891e+00
## 49850 -1.207509e+00 -1.207467e+00
## 49851 -1.200896e+00 -1.200854e+00
## 49852 -1.194093e+00 -1.194053e+00
## 49853 -1.187102e+00 -1.187062e+00
## 49854 -1.179922e+00 -1.179883e+00
## 49855 -1.172553e+00 -1.172515e+00
## 49856 -1.164997e+00 -1.164960e+00
## 49857 -1.157253e+00 -1.157216e+00
## 49858 -1.149320e+00 -1.149285e+00
## 49859 -1.141201e+00 -1.141167e+00
## 49860 -1.132896e+00 -1.132862e+00
## 49861 -1.124406e+00 -1.124373e+00
## 49862 -1.115732e+00 -1.115700e+00
## 49863 -1.106875e+00 -1.106844e+00
## 49864 -1.097837e+00 -1.097807e+00
## 49865 -1.088618e+00 -1.088589e+00
## 49866 -1.079220e+00 -1.079191e+00
## 49867 -1.069643e+00 -1.069615e+00
## 49868 -1.059888e+00 -1.059861e+00
## 49869 -1.049958e+00 -1.049932e+00
## 49870 -1.039852e+00 -1.039827e+00
## 49871 -1.029572e+00 -1.029548e+00
## 49872 -1.019120e+00 -1.019096e+00
## 49873 -1.008495e+00 -1.008473e+00
## 49874 -9.976996e-01 -9.976781e-01
## 49875 -9.867344e-01 -9.867138e-01
## 49876 -9.756006e-01 -9.755809e-01
## 49877 -9.642993e-01 -9.642805e-01
## 49878 -9.528315e-01 -9.528136e-01
## 49879 -9.411983e-01 -9.411813e-01
## 49880 -9.294008e-01 -9.293847e-01
## 49881 -9.174402e-01 -9.174250e-01
## 49882 -9.053174e-01 -9.053031e-01
## 49883 -8.930336e-01 -8.930202e-01
## 49884 -8.805899e-01 -8.805774e-01
## 49885 -8.679873e-01 -8.679757e-01
## 49886 -8.552269e-01 -8.552163e-01
## 49887 -8.423100e-01 -8.423003e-01
## 49888 -8.292380e-01 -8.292292e-01
## 49889 -8.160126e-01 -8.160047e-01
## 49890 -8.026351e-01 -8.026281e-01
## 49891 -7.891073e-01 -7.891012e-01
## 49892 -7.754306e-01 -7.754254e-01
## 49893 -7.616066e-01 -7.616024e-01
## 49894 -7.476368e-01 -7.476335e-01
## 49895 -7.335228e-01 -7.335205e-01
## 49896 -7.192662e-01 -7.192648e-01
## 49897 -7.048685e-01 -7.048680e-01
## 49898 -6.903312e-01 -6.903316e-01
## 49899 -6.756559e-01 -6.756573e-01
## 49900 -6.608442e-01 -6.608465e-01
## 49901 -6.458975e-01 -6.459008e-01
## 49902 -6.308176e-01 -6.308217e-01
## 49903 -6.156058e-01 -6.156109e-01
## 49904 -6.002637e-01 -6.002698e-01
## 49905 -5.847930e-01 -5.848000e-01
## 49906 -5.691951e-01 -5.692030e-01
## 49907 -5.534716e-01 -5.534805e-01
## 49908 -5.376241e-01 -5.376339e-01
## 49909 -5.216540e-01 -5.216648e-01
## 49910 -5.055631e-01 -5.055747e-01
## 49911 -4.893527e-01 -4.893653e-01
## 49912 -4.730245e-01 -4.730380e-01
## 49913 -4.565800e-01 -4.565944e-01
## 49914 -4.400207e-01 -4.400361e-01
## 49915 -4.233483e-01 -4.233647e-01
## 49916 -4.065647e-01 -4.065820e-01
## 49917 -3.896721e-01 -3.896903e-01
## 49918 -3.726724e-01 -3.726916e-01
## 49919 -3.555679e-01 -3.555880e-01
## 49920 -3.383606e-01 -3.383817e-01
## 49921 -3.210526e-01 -3.210746e-01
## 49922 -3.036460e-01 -3.036689e-01
## 49923 -2.861429e-01 -2.861668e-01
## 49924 -2.685454e-01 -2.685702e-01
## 49925 -2.508556e-01 -2.508813e-01
## 49926 -2.330755e-01 -2.331022e-01
## 49927 -2.152074e-01 -2.152350e-01
## 49928 -1.972532e-01 -1.972817e-01
## 49929 -1.792151e-01 -1.792445e-01
## 49930 -1.610952e-01 -1.611255e-01
## 49931 -1.428955e-01 -1.429268e-01
## 49932 -1.246182e-01 -1.246504e-01
## 49933 -1.062654e-01 -1.062985e-01
## 49934 -8.783913e-02 -8.787317e-02
## 49935 -6.934150e-02 -6.937646e-02
## 49936 -5.077462e-02 -5.081050e-02
## 49937 -3.214059e-02 -3.217738e-02
## 49938 -1.344150e-02 -1.347920e-02
## 49939 5.320558e-03 5.281949e-03
## 49940 2.414348e-02 2.410396e-02
## 49941 4.302517e-02 4.298475e-02
## 49942 6.196354e-02 6.192221e-02
## 49943 8.095647e-02 8.091425e-02
## 49944 1.000017e-01 9.995861e-02
## 49945 1.190970e-01 1.190529e-01
## 49946 1.382398e-01 1.381949e-01
## 49947 1.574278e-01 1.573820e-01
## 49948 1.766588e-01 1.766121e-01
## 49949 1.959302e-01 1.958826e-01
## 49950 2.152398e-01 2.151913e-01
## 49951 2.345851e-01 2.345358e-01
## 49952 2.539639e-01 2.539137e-01
## 49953 2.733737e-01 2.733226e-01
## 49954 2.928122e-01 2.927602e-01
## 49955 3.122770e-01 3.122242e-01
## 49956 3.317658e-01 3.317121e-01
## 49957 3.512761e-01 3.512216e-01
## 49958 3.708057e-01 3.707503e-01
## 49959 3.903521e-01 3.902959e-01
## 49960 4.099131e-01 4.098560e-01
## 49961 4.294861e-01 4.294282e-01
## 49962 4.490689e-01 4.490102e-01
## 49963 4.686591e-01 4.685995e-01
## 49964 4.882544e-01 4.881940e-01
## 49965 5.078523e-01 5.077911e-01
## 49966 5.274506e-01 5.273885e-01
## 49967 5.470467e-01 5.469838e-01
## 49968 5.666385e-01 5.665748e-01
## 49969 5.862234e-01 5.861589e-01
## 49970 6.057992e-01 6.057339e-01
## 49971 6.253635e-01 6.252974e-01
## 49972 6.449138e-01 6.448469e-01
## 49973 6.644474e-01 6.643797e-01
## 49974 6.839617e-01 6.838932e-01
## 49975 7.034539e-01 7.033847e-01
## 49976 7.229214e-01 7.228514e-01
## 49977 7.423615e-01 7.422907e-01
## 49978 7.617714e-01 7.616998e-01
## 49979 7.811486e-01 7.810762e-01
## 49980 8.004902e-01 8.004171e-01
## 49981 8.197937e-01 8.197198e-01
## 49982 8.390563e-01 8.389817e-01
## 49983 8.582753e-01 8.582000e-01
## 49984 8.774480e-01 8.773720e-01
## 49985 8.965719e-01 8.964951e-01
## 49986 9.156441e-01 9.155666e-01
## 49987 9.346619e-01 9.345837e-01
## 49988 9.536228e-01 9.535439e-01
## 49989 9.725239e-01 9.724443e-01
## 49990 9.913627e-01 9.912824e-01
## 49991 1.010136e+00 1.010055e+00
## 49992 1.028842e+00 1.028761e+00
## 49993 1.047478e+00 1.047395e+00
## 49994 1.066040e+00 1.065957e+00
## 49995 1.084526e+00 1.084443e+00
## 49996 1.102934e+00 1.102850e+00
## 49997 1.121261e+00 1.121176e+00
## 49998 1.139504e+00 1.139418e+00
## 49999 1.157660e+00 1.157574e+00
## [ reached 'max' / getOption("max.print") -- omitted 70001 rows ]
\[g(E(Y_i(t))) = \beta_0+\beta_1t+b_{i0}+b_{i1}t\] - But this model is obviously too simple for the data generation scheme
# model fit
t1 <- Sys.time()
fit_adglmm <- mixed_model(Y ~ t, random = ~ t | id, data = train_df, family = binomial())
t2 <- Sys.time()
# time
t2-t1
## Time difference of 2.23976 mins
# model
summary(fit_adglmm)
##
## Call:
## mixed_model(fixed = Y ~ t, random = ~t | id, data = train_df,
## family = binomial())
##
## Data Descriptives:
## Number of Observations: 500000
## Number of Groups: 500
##
## Model:
## family: binomial
## link: logit
##
## Fit statistics:
## log.Lik AIC BIC
## -324123.9 648257.7 648278.8
##
## Random effects covariance matrix:
## StdDev Corr
## (Intercept) 1.1840
## t 2.3831 -0.9987
##
## Fixed effects:
## Estimate Std.Err z-value p-value
## (Intercept) 0.0615 0.0533 1.1546 0.24827
## t -0.1185 0.1071 -1.1059 0.26877
##
## Integration:
## method: adaptive Gauss-Hermite quadrature rule
## quadrature points: 11
##
## Optimization:
## method: EM
## converged: TRUE
\[g(E(Y_i(t))) = \sum_{k=1}^5\zeta_{k}B_k(t)+\sum_{l=1}^4\xi_{il}\phi_l(t)\]
# subset 10 subjects from training dataset
df_train_sub <- train_df %>% filter(id %in% 1:10 & sind==bin)
df_train_sub$id <- droplevels(df_train_sub$id)
# levels(df_train_sub$id)
# intersect(t, unique(df_train_sub$t))
PS 50 subjects took 1.06 hours to fit with the same model structure.
# construct design matrix for the fixed effect intercept
t_sub <- unique(df_train_sub$t)
fix_xmat <- smoothCon(s(t, bs="cr", k=5), data = data.frame(t=t_sub))
dim(fix_xmat[[1]]$X)
## [1] 100 5
# basis functions
fix_xmat <- data.frame(t=t_sub,
fix_xmat[[1]]$X)
colnames(fix_xmat) <- c("t", paste0("B", 1:5))
df_train_sub <- df_train_sub %>% left_join(fix_xmat, by = "t")
# construct design matrix for the random effect
rand_xmat <- smoothCon(s(t, bs="cr", k=4), data = data.frame(t=t_sub))
dim(rand_xmat[[1]]$X) # 5 basis functions for each of the 50 subjects
## [1] 100 4
rand_xmat <- data.frame(t = t, rand_xmat[[1]]$X)
df_train_sub <- df_train_sub %>%
left_join(rand_xmat, by = c("t"))
The model fitting took about 6 minutes at this data scale, which obviously is not realistic for 500 simulated datasets.
Also tried increasing number of basis functions, which ended up with “computationally singular”. In fact, this method does not seem very numerically stable.
# model fit
t1 <- Sys.time()
fit_adglmm <- mixed_model(fixed = Y ~ 0+B1+B2+B3+B4+B5,
random = ~X1+X2+X3+X4 | id,
data = df_train_sub, family = binomial())
t2 <- Sys.time()
t2-t1
## Time difference of 5.761589 mins
# model
summary(fit_adglmm)
##
## Call:
## mixed_model(fixed = Y ~ 0 + B1 + B2 + B3 + B4 + B5, random = ~X1 +
## X2 + X3 + X4 | id, data = df_train_sub, family = binomial())
##
## Data Descriptives:
## Number of Observations: 1000
## Number of Groups: 10
##
## Model:
## family: binomial
## link: logit
##
## Fit statistics:
## log.Lik AIC BIC
## -692.0652 1424.13 1430.182
##
## Random effects covariance matrix:
## StdDev Corr
## (Intercept) 0.4447 (Intr) X1 X2 X3
## X1 0.4525 -0.9789
## X2 0.4473 -0.9889 0.9535
## X3 0.4468 -0.9906 0.9800 0.9755
## X4 0.4603 -0.9722 0.9478 0.9499 0.9514
##
## Fixed effects:
## Estimate Std.Err z-value p-value
## B1 0.3231 0.2490 1.2977 0.19441
## B2 -0.0915 0.1407 -0.6505 0.51534
## B3 -0.0326 0.1397 -0.2337 0.81520
## B4 -0.0570 0.1406 -0.4052 0.68530
## B5 0.1543 0.2476 0.6230 0.53326
##
## Integration:
## method: adaptive Gauss-Hermite quadrature rule
## quadrature points: 7
##
## Optimization:
## method: hybrid EM and quasi-Newton
## converged: TRUE
df_test_sub <- test_df %>% filter(id %in% 1:10 & sind==bin)
df_test_sub$id <- droplevels(df_test_sub$id)
# levels(df_test_sub$id)
# basis functions for both fixed and random effects should be shared across sample
df_test_sub <- df_test_sub %>% left_join(fix_xmat, by = "t") %>%
left_join(rand_xmat, by = "t")
# given observation from 0.2, 0.4, 0.6, 0.8, predict the rest
# this step in fact is very fast
pred_adglmm1 <- predict(fit_adglmm,
newdata = df_test_sub %>% filter(t<=0.2),
newdata2 = df_test_sub %>% filter(t>0.2),
type = "subject_specific", type_pred = "link",
return_newdata = TRUE)
pred_adglmm2 <- predict(fit_adglmm,
newdata = df_test_sub %>% filter(t<=0.4),
newdata2 = df_test_sub %>% filter(t>0.4),
type = "subject_specific", type_pred = "link",
return_newdata = TRUE)
pred_adglmm3 <- predict(fit_adglmm,
newdata = df_test_sub %>% filter(t<=0.6),
newdata2 = df_test_sub %>% filter(t>0.6),
type = "subject_specific", type_pred = "link",
return_newdata = TRUE)
pred_adglmm4 <- predict(fit_adglmm,
newdata = df_test_sub %>% filter(t<=0.8),
newdata2 = df_test_sub %>% filter(t>0.8),
type = "subject_specific", type_pred = "link",
return_newdata = TRUE)
pred_adglmm_all <- df_test_sub %>%
left_join(pred_adglmm1$newdata2 %>% select(id, t, pred), by = c("id", "t")) %>%
left_join(pred_adglmm2$newdata2 %>% select(id, t, pred), by = c("id", "t")) %>%
left_join(pred_adglmm3$newdata2 %>% select(id, t, pred), by = c("id", "t")) %>%
left_join(pred_adglmm4$newdata2 %>% select(id, t, pred), by = c("id", "t"))
colnames(pred_adglmm_all) <- c(colnames(pred_adglmm_all)[1:15], "pred1", "pred2", "pred3", "pred4")
pred_adglmm_all %>%
filter(id %in% 1:4) %>%
mutate_at(vars(eta_i, pred1, pred2, pred3, pred4), function(x){exp(x)/(1+exp(x))}) %>%
ggplot()+
geom_point(aes(x=t, y=Y), size = 0.2)+
geom_line(aes(x=t, y=eta_i, col = "True"))+
geom_line(aes(x=t, y=pred1, col = "0.2"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred2, col = "0.4"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred3, col = "0.5"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred4, col = "0.8"), linetype="dashed", na.rm=T)+
facet_wrap(~id)+
labs(title = "GLMMadaptive + spline basis prediction")